In classical physics if we have two distinct physical states ψ1 6=ψ2 of a physical system and we know that ψ1ORψ2 is a true statement we can easily deduce that ψ1XORψ2 is a true statement too. In Quantum Mechanics however we encounter a novel possibility for quantum coherent superposition. It has been verified in numerous experiments that a qubit can be prepared in a linear combination of two orthogonal states, and this parallel existence in the quantum realm, in the form ψ1ANDψ2, is what requires caution when we draw conclusions from a given set of premises — the truth of ψ1ORψ2 now does not lead to the truth of ψ1XORψ2. If a qubit at point x is in a state ψ1XORψ2 then ψ1 and ψ2 are called distinguishable states. Logically, if the qubit at point x is in a state ψ1XNORψ2 the two states ψ1 and ψ2 will be indistinguishable. From the requirement for mathematical consistency it follows that two states ψ1 and ψ2 cannot be both distinguishable and indistinguishable at the same time. The concept of distinguishability is intimately tied with the notion of quantum complementarity. While the quantum amplitudes evolve linearly according to the Schrodinger equation, the physical observables are obtained from the underlying quantum amplitudes through nonlinearity prescribed by Born’s rule. Thus if quantum states ψ1(x) 6=0 and ψ2(x) 6=0 are indistinguishable at a point x (coherent superposition), that is ψ1(x)ANDψ2(x), the probability distribution (observed intensity) is given by P = |ψ1(x) + ψ2(x)|. The density matrix of the setup is a pure type one, ρ= ( |ψ1| 2 ψ1ψ ∗ 2
In classical physics if we have two distinct physical states ψ1 6=ψ2 of a physical system and we know that ψ1ORψ2 is a true statement we can easily deduce that ψ1XORψ2 is a true statement too. In Quantum Mechanics however we encounter a novel possibility for quantum coherent superposition. It has been verified in numerous experiments that a qubit can be prepared in a linear combination of two orthogonal states, and this parallel existence in the quantum realm, in the form ψ1ANDψ2, is what requires caution when we draw conclusions from a given set of premises — the truth of ψ1ORψ2 now does not lead to the truth of ψ1XORψ2. If a qubit at point x is in a state ψ1XORψ2 then ψ1 and ψ2 are called distinguishable states. Logically, if the qubit at point x is in a state ψ1XNORψ2 the two states ψ1 and ψ2 will be indistinguishable. From the requirement for mathematical consistency it follows that two states ψ1 and ψ2 cannot be both distinguishable and indistinguishable at the same time. The concept of distinguishability is intimately tied with the notion of quantum complementarity. While the quantum amplitudes evolve linearly according to the Schrodinger equation, the physical observables are obtained from the underlying quantum amplitudes through nonlinearity prescribed by Born’s rule. Thus if quantum states ψ1(x) 6=0 and ψ2(x) 6=0 are indistinguishable at a point x (coherent superposition), that is ψ1(x)ANDψ2(x), the probability distribution (observed intensity) is given by P = |ψ1(x) + ψ2(x)|. The density matrix of the setup is a pure type one, ρ= ( |ψ1| 2 ψ1ψ ∗ 2
For around five decades, physicists have been experimenting with single quanta such as single photons. Insofar as the practised ensemble reasoning has become obsolete for the interpretation of these experiments, the non-classical intrinsic probabilistic nature of quantum theory has gained increased importance. One of the most important exclusive features of quantum physics is the undeniable existence of the superposition of states, even for single quantum objects. One known example of this effect is entanglement. In this paper, two classically contradictory phenomena are combined to one single experiment. This experiment incontestably shows that a single photon incident on an optical beam splitter can either be reflected or transmitted. The almost complete absence of coincident clicks of two photodetectors demonstrates that these two output states are incompatible. However, when combining these states using two mirrors, we can observe interference patterns in the counting rate of the single photon detector. The only explanation for this is that the two incompatible output states are prepared and kept simultaneously—a typical consequence of a quantum superposition of states. (Semi-)classical physical concepts fail here, and a full quantum concept is predestined to explain the complementary experimental outcomes for the quantum optical “non-waves” called single photons. In this paper, we intend to demonstrate that a true quantum physical key experiment (“true” in the sense that it cannot be explained by any classical physical concept), when combined with full quantum reasoning (probability, superposition and interference), influences students’ readiness to use quantum elements for interpretation.
We discuss the multiple pass interferometer setup proposed by Unruh, and clarify some of the fundamental issues linked with complementarity. We explicitly state all mathematical instructions for manipulating the quantum amplitudes and assessing the probability distribution functions. In this respect we show that certain purely math logical limitations (requirement for consistency) prevent one to argue that there is one-to-one corespondence between paths 1 and 2 and the exit gates 10 and 9 ("which way" interpretation), and at the same time insist on pure state density matrix, i. e. existent nonmeasured interference in the second building block of Unruh's interferometer. Furthermore one cannot even argue that Unruh's setup is described by mixed density matrix that keeps the one-to-one correspondence between the paths 1 and 2 and the exit gates. This last claim is mathematically consistent, however is experimentally disprovable - because one may potentially distinguish mixed quantum state from pure quantum state. One just lets the two beams captured at the exit gates cross each other. If interference can be observed the two exit gates were coherent and provide beams in pure state (superposition), while if interference cannot be observed, the state of the exit gates was mixed one. Since the captured beams at the exit gates in Unruh's experiment could interfere this implies that the whole setup is characterized with pure state density matrix and does not preserve the one-to-one correspondence between the entry points and exit gates, even in case where the destructive interference in arm 5 of the interferometer is not measured. Therefore the correct (experimentally plausible and mathematically consistent) exposition of complementarity introduced by Georgiev in 2004 is that Unruh's setup is characterized by pure state density matrix and does not keep the suggested by Unruh one-to-one correspondence. As an appendix is shown the equivalence between Unruh's setup and Afshar's setup and correct analysis of Afshar's experiment is also provided.
Two photons are said to be identical if they are prepared in the same quantum state. Given the latter, there is a unique way to achieve this. Conversely, there are many different ways of preparing two non-identical photons: they may differ in frequency, polarization, amplitude, etc. Therefore, photon distinguishability depends upon the specific degree of freedom being varied. By means of a careful analysis of the coincidence probability distribution in a Hong–Ou–Mandel experiment, we can show that photon distinguishability can be actually quantified by the rate of distinguishability of photons, an experimentally measurable parameter that crucially depends on both the photon quantum state and the degree of freedom under control.
According to a recent no-go theorem [M. Pusey, J. Barrett and T. Rudolph, Nat. Phys. 8, 475 (2012)], models in which quantum states correspond to probability distributions over the values of some underlying physical variables must have the following feature: the distributions corresponding to distinct quantum states do not overlap. In such a model, it cannot coherently be maintained that the quantum state merely encodes information about underlying physical variables. The theorem, however, considers only models in which the physical variables corresponding to independently prepared systems are independent, and this has been used to challenge the conclusions of that work. Here we consider models that are defined for a single quantum system of dimension d, such that the independence condition does not arise, and derive an upper bound on the extent to which the probability distributions can overlap. In particular, models in which the quantum overlap between pure states is equal to the classical overlap between the corresponding probability distributions cannot reproduce the quantum predictions in any dimension d ≥ 3. Thus any ontological model for quantum theory must postulate some extra principle, such as a limitation on the measurability of physical variables, to explain the indistinguishability of quantum states. Moreover, we show that as d→∞, the ratio of classical and quantum overlaps goes to zero for a class of states. The result is noise tolerant, and an experiment is motivated to distinguish the class of models ruled out from quantum theory.